The sum and difference of the LCM and the HCF of the two numbers are 624 and 546 respectively. If the sum of two numbers is 312, find the numbers.a)39,273b)117,195c)Data inadequated)None of theseCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Let the LCM and HCF are p and q, respectively.

So, p + q = 624, p - q = 546

Therefore, p = (624+546)/2 = 585 and q = (624-546)/2 = 39

That means LCM = 585 and HCF = 39

Now, let the numbers be 39a and 39b, where a and b are co-primes.

Therefore 39a + 39b = 312 => a + b = 312/39 = 8

So possible pairs of co-primes, whose sum is 8, are (3,5) and (1,7).

Therefore possible pairs of numbers are

(39*3, 39*5) = (117,195) and

(39*1, 39*7) = (39,273)

Now, LCM*HCF = 585*39 = 22815

Also, 117*195 = 22815 and 39*273 = 10647

Hence, the required numbers are 117 and 195 because the product of two numbers is equal to the product of their LCM and HCF.

Answer

To solve this problem, let's assume the two numbers as a and b.

Let's break down the given information step by step:

1. The sum of the two numbers is 312:
a + b = 312

2. The sum of the LCM (Least Common Multiple) and the HCF (Highest Common Factor) of the two numbers is 624:
LCM(a, b) + HCF(a, b) = 624

3. The difference between the LCM and the HCF of the two numbers is 546:
LCM(a, b) - HCF(a, b) = 546

Now, let's solve these equations to find the values of a and b:

1. Subtracting equation 2 from equation 3, we get:
(LCM(a, b) + HCF(a, b)) - (LCM(a, b) - HCF(a, b)) = 624 - 546
2 * HCF(a, b) = 78
HCF(a, b) = 39

2. Substituting the value of HCF in equation 2, we get:
LCM(a, b) + 39 = 624
LCM(a, b) = 624 - 39
LCM(a, b) = 585

3. Now, we need to find two numbers whose LCM is 585 and HCF is 39. Let's try some numbers:

- If we take a = 117 and b = 195, then LCM(117, 195) = 585 and HCF(117, 195) = 39.
- If we take a = 39 and b = 273, then LCM(39, 273) = 585 and HCF(39, 273) = 39.

Both sets of numbers satisfy the given conditions, but we need to find the sum of the two numbers, which is 312. Only the numbers a = 117 and b = 195 satisfy this condition.

Therefore, the numbers are 117 and 195, which is option B.

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